RU2527141C1 - Method of expanding measurement range of angular velocities of closed feedback loop fibre-optic gyroscope - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: method includes performing modulation with amplitude of 0, ±π rad and generating initial phase shift between beams of a fibre ring interferometer (FRI), which is equal to ±π/2 rad, using stepped triangular sawtooth voltage. When changing the Sagnac phase difference by changing the frequency of the stepped sawtooth voltage and changing polarity of connection of electrodes of the phase modulator of the FRI, the phase difference of the beams thereof assumes the following discrete series of values: Φ_s+Ψ_stv=±2(n+1)/2×π rad, where n=0, 1, 2, …, Φ_s is the Sagnac phase difference, and Ψ_stv is the phase difference of the FRI beams by applying a stepped triangular sawtooth voltage across the phase modulator.

EFFECT: wider measurement range of angular velocities of a closed feedback loop fibre-optic gyroscope.

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Description

The invention relates to the field of fiber optics and can be used in the design of fiber-optic gyroscopes and other sensors of physical quantities based on single-mode optical fibers.

The fiber optic gyroscope (hereinafter referred to as FOG) contains a fiber ring interferometer (FRI) and an electronic information processing unit. The FRI contains a source of optical radiation, a fiber divider of radiation power, an integrated optical circuit (hereinafter - IOS), a multi-turn sensitive coil and a photodetector. IOS contains in its composition a Y-divider of the power of optical radiation based on polarizing channel waveguides and a phase modulator located on the output arms of the Y-divider. Channel waveguides of the Y-divider are formed in a lithium niobate substrate using proton-exchange technology, which allows the waveguides to acquire polarizing properties. On the output channel waveguides is a phase modulator of optical rays passing through the channel waveguides. The phase modulator is a metal electrode deposited on both sides of the channel waveguides. When an electric voltage is applied to the electrodes due to the electro-optical effect in the material of the channel waveguides, the refractive index changes, which leads to the effect of phase modulation of the optical rays propagating along the channel waveguides. The ends of the optical fibers of the sensitive gyro coil are docked to the output waveguides of the Y-divider.

An interference pattern is formed on the FRI photodetector, formed by two optical beams that have passed the sensitive coil of the gyroscope in two mutually opposite directions. When the ring interferometer rotates between these two beams, due to the Sagnac effect, a phase difference arises, which is expressed as follows:

Φ _c = [4πRL / λc] × Ω,

where R is the radius of the sensitive coil of the gyroscope;

L is the fiber length of the coil;

λ is the central wavelength of the radiation source;

c is the speed of light in vacuum;

Ω is the angular velocity of rotation of the gyroscope.

Thus, the optical radiation power at the photodetector can be represented as

P _f = ½P ₀ (1 + cosΩ _c ),

where P ₀ is the power of the rays interfering at the photodetector.

To increase the sensitivity of VOG near zero angular velocities, auxiliary phase modulation is used. To achieve the effect of phase modulation of rays in a ring interferometer using an IOS phase modulator, we use the time delay of the ray fronts interfering on the photodetector while passing through the IOS phase modulator. This time delay is equal to the travel time of the FRI light rays along the fiber of the sensitive coil and is

$$\tau =\frac{L{n}_0}{c}$$

where n ₀ is the refractive index of the material of the fiber of the sensitive coil.

When applying voltage pulses to the phase modulator that follow with a frequency of 1 / 2τ and introduce the phase difference between the FRI beams in the form of a pulse sequence with amplitudes ± π / 2 radians ± 3π / 2 radians, the photodetector current can be represented as

I _Ф = ½Р ₀ η _ф (1 + cosθ _m cosФ _c + sinθ _m sinФ _c ),

where η _f is the current sensitivity of the photodetector,

θ _m is the amplitude of the auxiliary phase modulation.

Next, the signal from the photodetector is fed to the input of the photodetector current amplifier, at the output of which there is a voltage proportional to

U = ½P ₀ η _f R _H (1 + cosθ _m cos _{c c} + sinθ _m sin _{c c} ),

where R _n - load resistance of the current amplifier of the photodetector.

In [1], a method for linearizing the output characteristic of VOG is proposed. At the same time as the auxiliary phase modulation voltage, a step-like sawtooth voltage is applied to the phase modulator to compensate for the Sagnac phase difference. The work of VOG is described in detail in [2]. Using a sawtooth voltage supplied to the phase modulator, a controlled phase difference between the FRI beams is introduced, with which the Sanka phase difference is compensated. For this purpose, a closed feedback loop (FOG with a closed feedback loop) is organized to reset the signal at the output of the synchronous gyro rotation signal detector. The signal at the output of the synchronous detector is automatically reset by selecting the voltage step value of the step-like sawtooth voltage (SPN). Due to this, the output characteristic of the VOG becomes linear. The signal at the output of the photodetector current amplifier in this case can be represented as

U = ½P ₀ η _Ф R _n [± sinθ _m (Ф _с -Ψ _к )],

where Ψ _k is the adjustable phase difference, which is introduced between the FRI beams using the SPN when applying it to the phase modulator.

For the frequency of SPN in this case, the following relation is true:

$$f_n(t)=\frac{\mathrm{four}\pi RL}{\lambda c}\cdot \frac{\mathrm{one}}{\eta U_P\tau ct}\Omega (t)$$

where R is the radius of the sensitive coil, L is the fiber length of the sensitive coil, λ is the radiation wavelength of the source, s is the speed of light in vacuum, η is the efficiency of the phase modulator, U _p is the amplitude of the SPN, τ _st is the travel time of the light rays along the sensitive fiber coils, Ω (t) is the angular velocity of rotation.

The FOG scale factor is stabilized by providing the amplitude of the SPD, which, when applied to the phase modulator, changes the phase of the FRI beams by 2 π radians. The amplitude of the SPN is regulated by isolating the pulse of illumination of the photodetector when resetting the voltage of the SPN and then resetting it by adjusting the amplitude of the SPN. In this case, for the frequency of SPN the following relation is true:

f _n (t) = [2R / λn] × Ω (t),

where n is the refractive index of the fiber material.

Thus, the FOG scale factor does not depend on the efficiency of the phase modulator, which has great instability when exposed to external destabilizing factors. In order of magnitude, the stability of the scale factor is most affected by ηU _П due to the large instability of the efficiency η of the phase modulator. Therefore, in order to stabilize this value, and ultimately stabilize the scale factor, in the information processing scheme with a closed feedback loop, it is necessary to ensure the amplitude of the SPD, which introduces a phase difference between the FRI rays equal to 2π radians. If this condition is not met, then when the SPN is reset, a light pulse appears on the photodetector, which is then amplified by the current amplifier of the photodetector. Then, the amplitude of the SPD is changed so that when pulsing the SPT at the photodetector, light pulses do not occur [1]. At low angular velocities, information on adjusting the amplitude of the SPN is rarely received, since the period of the SPN depends on the angular velocity, and therefore the known method of stabilization of the scale factor does not provide the necessary stability at small angular velocities.

A known method of stabilization of the scale factor [3] due to the organization of the second feedback loop. To do this, using an additional second demodulator, a mismatch signal is generated, which is generated on the photodetector when the efficiency of the phase modulator of the IOS FRI changes. Next, the signal from the output of the second demodulator enters the second regulator, which stabilizes the amplitude of the auxiliary phase modulation (VFM) using the second regulator by changing the voltage of the VFM. The stabilization of the amplitude of the WFM is achieved at a zero level of the error signal at the output of the second demodulator. The error signal is high-frequency and does not depend on the angular velocity. Thus, an increase in the stability of the scale factor of the gyroscope is achieved even at low angular velocities by increasing the speed of the second feedback loop. The SPN amplitude code is connected to the VFM voltage amplitude code by a hard numerical ratio and therefore is set automatically.

In [5], the use of triangular-shaped SPNs is considered. When using the operation of switching the polarity of the electrodes of the phase modulator at the times of the minimum and maximum voltage values (the maximum level of a triangular SPN when applied to a phase modulator changes the phase of the FRI rays to the n radian), the triangular SNP is transformed into a SPN either with increasing or only with decreasing fronts and with an amplitude that changes the phase of the FRI rays by 2π radians.

A disadvantage of the known circuit [1, 2] is the presence of a deadband in the VOG, which is determined by electrical pickups, a spurious Michelson interferometer arising due to back reflections in the IOS [4], and the instability of the time parameters of the auxiliary phase modulation voltage (VFM). The instability of the time parameters lies mainly in the instability of the phase of the voltage of the WFM.

радиан и $${\Psi }_{п}^2=(2\pi -2\Delta )$$

радиан. При использовании СПН с амплитудой $${\Psi }_{п}^1=2\Delta$$

радиан сброс максимального напряжения СПН необходимо производить в течение отрицательного полупериода сигнала вращения при положительной угловой скорости и в течение положительного полупериода сигнала вращения при отрицательной угловой скорости. При использовании этого вида СПН максимальный диапазон компенсации разности фаз Саньяка составляет Δ радиан, что приводит к сужению диапазона измеряемых угловых скоростей по сравнению с ВОГ, предложенным в [1, 2]. Таким образом, при устранении зоны нечувствительности ВОГ происходит сужение диапазона измеряемых угловых скоростей, что является недостатком ВОГ, предложенного в [3, 4]. Максимальный диапазон измеряемых угловых скоростей достигается при Δ=π/2 радиан, при этом амплитуда периодической последовательности импульсов ВФМ составляет ±π/2 радиан и ±3π/2 радиан. Для устранения зоны нечувствительности ВОГ в этом случае величина $${\Psi }_{п}^1=\pi$$

радиан, что не дает возможности обеспечить временную стабильность фазы импульсной последовательности ВФМ. Для устранения этого недостатка используется СПН треугольной формы с амплитудой $${\Psi }_{п}^1=\pi /2$$

радиан, при этом необходимо использовать смену полярности подключения электродов фазового модулятора. Но даже в этом случае максимальный диапазон компенсируемой разности фаз Саньяка Ф_с составляет величину ±π/4 радиан [5].A known method of eliminating the dead zone of VOG [4, 6, 7]. The WFM is carried out in the form of a periodic sequence of rectangular pulses with amplitudes ± (π + Δ) radians and ± (π + Δ) radians, where Δ = π / 2 ⁿ and n = 1, 2, 3, .... The voltage waveform of the VFM is such that even if there is electrical interference, there is no spurious signal at the frequency of the VOG rotation signal at the output of its demodulator. When the rays of the parasitic Michelson interferometer are modulated when a WFM voltage is applied to the modulator, a parasitic signal also does not occur at the frequency of the rotation signal. But the main factor in the absence of a dead zone is the formation of a WFM voltage with time-stable parameters. By parameters that are constant over time are meant the temporary stability of the phase of the WFM voltage. But this is possible if the amplitude of the SPN is such that when it is applied to the phase modulator, the phase of the FRI rays does not change by the 2nd radian. The phase difference between the FRI rays when resetting the maximum SPN can take two values, namely $${\Psi }_P^{\mathrm{one}}=2\Delta$$

radian and $${\mathrm{<figure-callout\; id="2"\; label="\Psi \; P"\; filenames="00000013.png,00000014.png"\; state="\{\{state\}\}">\Psi \; </figure-callout>}}_P^{}=(2\pi -2\Delta )$$

radian. When using SPN with amplitude $${\Psi }_P^{\mathrm{one}}=2\Delta$$

radians, the reset of the maximum voltage of the SPN must be performed during the negative half-cycle of the rotation signal at a positive angular velocity and during the positive half-cycle of the rotation signal at a negative angular velocity. When using this type of SPN, the maximum compensation range for the Sagnac phase difference is Δ radian, which leads to a narrowing of the range of measured angular velocities in comparison with the FOG proposed in [1, 2]. Thus, when the VOG deadband is eliminated, the range of measured angular velocities narrows, which is a drawback of the VOG proposed in [3, 4]. The maximum range of measured angular velocities is achieved at Δ = π / 2 radians, while the amplitude of the periodic sequence of WFM pulses is ± π / 2 radians and ± 3π / 2 radians. To eliminate the dead zone of VOG in this case, the value $${\Psi }_P^{\mathrm{one}}=\pi$$

radian, which makes it impossible to provide temporary phase stability of the pulse sequence of the WFM. To eliminate this drawback, a triangular-shaped SPN with amplitude $${\Psi }_P^{\mathrm{one}}=\pi /2$$

radian, it is necessary to use a change in the polarity of the electrodes of the phase modulator. But even in this case, the maximum range of the compensated phase difference Sagnac Ф _с is ± π / 4 radians [5].

The aim of the present invention is to expand the range of measured angular velocities of the VOG, which eliminated the dead zone.

This goal is achieved by the fact that the phase difference between the beams of the ring interferometer is modulated in the form of a periodic sequence of rectangular pulses with a phase stable in time and with pulse amplitudes of 0, ± π radians, while ensuring the total difference of the phases of the rays by changing the frequency of the triangular step-like sawtooth voltage FRI equality _with F + Ψ _SPN = ± 2 (n + 1) / 2 × π radians, where n = 0, 1, 2, ..., Ψ _SPN - the phase difference between the beams of a ring interferometer, with the feed introduced on its phase modulator treug ceiling elements stepwise voltage.

The extension of the angular velocity measurement range is achieved through the use of modulation with an amplitude of 0, ± π radians and the formation of an initial phase shift between the FRI beams equal to ± π / 2 radians using a triangular sawtooth voltage (SPS) of a triangular shape. When changing the Sagnac phase difference by changing the frequency of the SPN and changing the polarity of the electrodes of the phase modulator of the fiber ring interferometer (FRI), the phase difference of its rays takes the following discrete series of values: Ф _s + Ψ _spn = ± 2 (n + 1) / 2 × π s (radian), where n = 0, 1, 2, ..., Ф _с is the Sagnac phase difference, and Ψ _spn = the phase difference of the FRI rays due to the supply of a triangular shape to the phase modulator of the SIC.

The invention is illustrated by drawings. Figure 1 shows the structural diagram of a fiber optic gyroscope with closed feedback loops. Figure 2 shows the voltage of the WFM and the phase difference of the FRI rays. Figure 3 shows the triangular shape SPN for the formation of the phase difference of the FRI rays ± π / 2 radians. Figure 4 shows the formation of the rotation signal at the photodetector. Figure 5 shows the formation of the error signal at the photodetector. Figure 6 shows the signal generation of the spurious Michelson interferometer at the photodetector. Figure 7 shows the phase change of the FRI rays to bring the FOG operating point to points with the phase difference of the rays (2n + 1) π / 2 radians.

The FRI VOG contains an optical radiation source 1 (FIG. 1) with a short coherence length, an optical beam splitter 2, an IOS 3, a fiber sensitive coil 4, and a photodetector 5. The IOS contains a Y-splitter of optical rays and two phase modulators formed on the output arms Y -divider by applying metal electrodes to the substrate. Channel waveguides of the Y-divider are formed in the IOS substrate from lithium niobate using the proton-exchange technology and therefore have polarizing properties. Thus, the IOS is a multifunctional element containing a polarizer, an optical beam splitter and two phase modulators. The light beam after the optical beam divider enters the IOS input and is divided into two beams by the Y-divider, which pass the sensing coil clockwise and counterclockwise and are again combined by the Y-divider into one beam, which, through the optical beams divider, enters the photodetector area. The optical radiation power at the photodetector can be represented as follows:

P _Φ = ½P ₀ [1 + cos Φ _c ].

Next, the signal is fed to the input of the current amplifier of the photodetector 6 and then to one of the two inputs of the differential amplifier 7. At its second input, a signal is received from a low-pass filter (integrator) 8, which eliminates the constant component of the signal at the photodetector at the output of the differential amplifier to ensure large gaining the variable component of the signal at the photodetector. Next, the signal from the output of the differential amplifier goes to the input of an analog-to-digital converter (ADC) 9 and then to the input of the digital information processing board 10. The signal after the ADC goes to the input of the synchronous detector (demodulator) of the VOG 11 rotation signal, then the signal from the output of the demodulator goes to input of the regulator 12 magnitude of the voltage step SPN. The control signal from the controller is fed to the input of the triangular SPN generator 13. The signal after the ADC is also fed to the second mismatch signal demodulator 14, after which it is fed to the input of the second controller15. The second regulator controls the magnitude of the amplitude of the VFM voltage, which is generated by the generator 16. The voltage from the voltage generator of the VFM and SPN is fed to the input of the adder 17, after which the total voltage is fed to the input of the digital-to-analog converter (DAC) 18 and then through the scaling operational amplifier 19 to the electrodes of the phase modulators IOS. The gyroscope rotation signal demodulator, the first voltage regulator of the SPN step, the SPS generator are part of the first VOG feedback loop for the rotation signal to be zero at the output of its demodulator by automatically selecting the SPN step using the regulator. The signal at the input of the rotation signal demodulator can be represented as

$${\text{U}}_{\text{d}}^{\text{one}}==\raisebox{1ex}{$\mathrm{one}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.P_o{\eta }_f{G}_0[\mathrm{one}+\cos {\theta }_m+\sin {\theta }_m+\sin (F_{\mathrm{from}}-{\Psi }_{\mathrm{from}Pn})]$$

where G ₀ is the gain of the photodetector current.

The voltage code of the SPN step is the output of the VOG. Using the first feedback loop, a compensation method is implemented for reading the phase difference of the FRI rays, which allows linearizing the output characteristic of the FOG.

The mismatch signal demodulator, the second VFM voltage amplitude regulator and the VFM voltage generator are part of the second feedback loop, which provides zeroing the mismatch signal at the output of the second demodulator by changing the VFM voltage amplitude using the regulator. Using the second feedback loop, the amplitude of the WFM is stabilized, and, as a consequence, the amplitude of the SPN as well. Thus, using the second feedback loop, an increase in the stability of the FOG scale factor is achieved.

Figure 2 shows the voltage of the VFM 20 and the phase difference between the beams of the FRI 21. The period of the voltage pulses is 4τ, where τ is the time equal to the travel time of the light waves along the sensitive coil of the FRI. The phase difference of the FRI rays during WFM is a periodic sequence of pulses with amplitudes of 0, ± π radians and a phase constant in time.

Figure 3 shows the triangular-shaped SNP for the formation of the phase difference of the FRI rays ± π / 2 radians. The voltage 24 has an amplitude, upon supply of which a phase difference π / 2 radians is introduced between the FRI beams on the IOS phase modulator. If, at time moments 22, 23 (voltage has a maximum value), the polarity of the connection of the electrodes of the IOS phase modulator is changed, and at time 24 (voltage has a minimum value), the polarity of the connection of the electrodes is restored, then the phase difference of the rays of the ring interferometer will change according to the law shown curve 25. This is due to the fact that when changing the direction of the electric field in the channel waveguide of the phase modulator, which is associated with a change in the polarity of the connection of its electric electrodes, the phase shift of the beam changes its sign. Thus, using the above SPN and the operation of changing the polarity of connecting the electrodes between the FRI beams, a constant shift of the phase difference equal to π / 2 radians is introduced. When changing the polarity of connecting the electrodes of the MOS phase modulator at times when the amplitude of the SPD has a minimum value, restoring the polarity of connecting the electrodes when the amplitude of the SPV has the maximum value, the law of phase change of the FRI rays looks like 26. In this case, a constant phase shift is introduced between the FRI beams, equal to -π / 2 radians.

Figure 4 shows the formation of the rotation signal at the photodetector. The signal is generated when the phase difference of the FRI rays is applied to the cosine curve 27, introduced by applying a triangular wave voltage to the phase-modulated IOS modulator, which introduces a constant phase difference shift equal to π / 2 radians. The Sagnac phase difference is shown by dashed line 28. As a result, a rotation signal 29 is generated on the photodetector. Thus, the signal at the input of the rotation signal demodulator can be represented as

$${\text{U}}_{\text{d}}^{\text{one}}==\raisebox{1ex}{$\mathrm{one}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.P_o{\eta }_f{G}_0[\mathrm{one}\pm \sin F_{\mathrm{from}}]$$

The stability of the initial shift of the phase difference of the FRI rays π / 2 radians is ensured by the operation of the second feedback loop.

Figure 5 shows the formation of the error signal at the photodetector. The dashed line 30 shows the change in the efficiency of the phase modulators of the IOS, and a mismatch signal 31 is generated on the photodetector. In the absence of rotation, the mismatch signal at the input of the second demodulator can be conditionally represented as

$${\text{<figure-callout id="2" label="U d" filenames="00000013.png,00000014.png" state="{{state}}">U </figure-callout>}}_{\text{d}}^{}==\raisebox{1ex}{$\mathrm{one}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.P_o{\eta }_f{G}_0[\mathrm{one}\pm \sin (U_{\pi }\Delta \eta ]$$

where Δη is the change in the efficiency of the IOS phase modulators, U _π is the voltage when applying to the electrodes of the IOS phase modulators the phase of the optical rays changes to π radian. As you can see, when detecting a rotation signal, there is no error due to the influence of the error signal, and, conversely, detection of the error signal does not contain errors due to the influence of the rotation signal.

In the well-known FOGs [1, 2], the existence of a dead zone is noted. As noted earlier, in addition to the instability of the WFM voltage phase, it can also arise due to electrical pickups from the WFM voltage supplied to the electrodes of the IOS phase modulators and from the existence of a spurious Michelson interferometer. The voltage of the VFM has such a shape that when signals are detected at the frequency of rotation signals and the error mismatch in demodulation, these signals do not occur.

Figure 6 shows the signal generation of the spurious Michelson interferometer at the photodetector. The Michelson spurious interferometer arises from the back reflections of radiation in the channel waveguides of the phase modulators of the IOS. Using the proposed WFM voltage, the signal of the parasitic interferometer has the form 32, and when this signal is detected at the frequency of the rotation signal, as well as at the frequency of the mismatch signal, it vanishes and the FOG deadband does not occur.

Figure 7 shows the phase change of the FRI rays to bring the FOG operating point to points with the total phase difference of the FRI rays (2n + 1) × π / 2 radians. At these points, as was shown above at the output of the demodulator of the rotation signal, it has a zero value, which makes it possible to implement the compensation method of reading the Sagnac phase difference. For example, as the positive angular velocity increases, the Sagnac phase difference begins to increase. In this case, the frequency of the triangular SPN 33 automatically changes from its maximum value to the minimum. The maximum value of the frequency of a triangular SPN is 1 / 4τ, and the minimum is determined by the capacity of the programmable logic integrated circuit (FPGA). Next, the corresponding mode of changing the polarity of the connection of the electrodes is selected in order to introduce a positive phase difference between the FRI beams in order to bring the total phase difference, i.e., the Sagnac phase difference and the phase difference introduced by the SPD to the phase difference of the rays equal to π / 2 radians. With Ф _с = π / 2 radians, the saw frequency has the lowest possible value. When exceeding the Sagnac phase difference F _with ≥π / 2 radians electrode polarity switching mode is changed so that the slope of the sawtooth law FRI ray phase change changed to the other 34, 35 and SPN frequency began to increase, and the phase difference due to the SPN is negative . At the maximum value of the SPN frequency, the Sagnac phase difference Φ _s = π radians. With a further increase in the Sagnac phase difference, the mode of switching the polarity of the electrodes again changes, but the SPN at the same time has a maximum frequency. This makes it possible to translate the VOG working point to a point with a total phase difference of FRI rays equal to 3π / 2 radians. Further, the process of changing the frequency of a triangular SPN is similar to how the Sagnac phase difference changed in the range from 0 to π radians. When using an operating point of 3π / 2 radians, the compensation method changes the Sagnac phase difference in the range from π to 2π radians. Thus, when operating points with a total phase difference of FRI rays of ± (2π + 1) × π / 2 radians are used, angular velocities are measured in a wide range, which is actually limited only by the coherence length of the radiation source.

Claims (1)

A method of expanding the range of measured angular velocities in a fiber-optic gyroscope with closed feedback loops, in which the auxiliary phase modulation voltage with time-stable parameters is used, as well as to compensate for the Sagnac Ф phase difference _{with a} triangular sawtooth voltage _{with a} step length of each step equal to the travel time of light rays through the fiber of the sensitive coil of the ring interferometer of a fiber-optic gyroscope, and with a voltage amplitude, and applying it to the phase modulator of the ring interferometer, the phase of the rays propagating through the sensitive coil changes by π radian, and the polarity of the electrodes of the phase modulator is changed, characterized in that the phase difference between the rays of the ring interferometer is modulated in the form of a periodic sequence of rectangular pulses with a stable in time phase and with pulse amplitudes of 0, ± π radians, while providing by changing the frequency of a triangular step sawtooth voltage of execution for the total phase difference beams fiber ring interferometer (FRI) _with the equality F + Ψ _SPN = ± 2 (n + 1) / 2 × π radians, where n = 0,1,2 ..., Ψ _SPN - the phase difference between rays of the ring interferometer introduced when a triangular step voltage is applied to its phase modulator.

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